We
study efficient and stable mechanisms in matching markets when the number of
agents is large and individuals’ preferences and priorities are drawn randomly.
When agents’ preferences are uncorrelated, then both efficiency and stability
can be achieved in an asymptotic sense via standard mechanisms such as deferred
acceptance and top trading cycles. When agents’ preferences are correlated over
objects, however, these mechanisms are either inefficient or unstable even in
an asymptotic sense. We propose a variant of deferred acceptance that is
asymptotically efficient, asymptotically stable and asymptotically incentive
compatible. This new mechanism performs well in a counterfactual calibration
based on New York City school choice data.

We
study top trading cycles in a two-sided matching environment (Abdulkadiroglu
and Sonmez (2003)) under the assumption that individuals’ preferences and
objects’ priorities are drawn iid uniformly. The distributions of agents’
preferences and objects’ priorities remaining after a given round of TTC depend
nontrivially on the exact history of the algorithm up to that round (and so
need not be uniform iid). Despite the nontrivial history-dependence of evolving
economies, we show that the number of individuals/objects assigned at each
round follows a simple Markov chain and we explicitly derive the transition
probabilities.

We
study Pareto efficient mechanisms in matching markets when the number of agents
is large and individual preferences are randomly drawn from a class of
distributions, allowing for both common and idiosyncratic shocks. We show that,
as the market grows large, all Pareto efficient mechanisms -- including top
trading cycles, serial dictatorship, and their randomized variants -- are
uniformly asymptotically payoff equivalent “up to the renaming of agents,”
yielding the utilitarian upper bound in the limit. This result implies that,
when the conditions of our model are met, policy makers need not discriminate
among Pareto efficient mechanisms based on the aggregate payoff distribution of
participants.